Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

Abstract

We calculate the level compressibility $\ensuremath{\chi}(W,L)$ of the energy levels inside $[\ensuremath{-}L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[\ensuremath{-}W/2,W/2]$. We show that $\ensuremath{\chi}(W,L)$ approaches the limit ${lim}_{L\ensuremath{\rightarrow}{0}^{+}}\ensuremath{\chi}(W,L)=0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of $W$ close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erd\"os-R\'enyi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.